Most related to our work, is the work by [1]} that
leveraged Euclidean optimal transport, parameterized using input
convex neural networks (ICNNs) [2]} to construct universal
normalizing flows on Euclidean spaces. Similarly, [3]}, [4]} compute optimal transport maps via ICNNs. Riemannian optimal transport replaces
the standard Euclidean convex functions with so-called \(c\) -convex or
\(c\) -concave functions, and the Euclidean translation by exponential
map. Unfortunately, the notion of \(c\) -convex or \(c\) -concave functions
is intricate and a simple characterization of such functions is not
known.
Our approach is to approximate arbitrary \(c\) -concave functions on
general Riemannian manifolds using discrete \(c\) -concave
functions that are simply the minimum of a finite number of
translated squared intrinsic distance functions, see
fig:discretecconcave. Intuitively, this construction resembles
the approximation of a Euclidean concave function as the minimum of a
finite collection of affine tangents. Although simple, we prove that
discrete \(c\) -concave functions are in fact dense in the
space of \(c\) -concave functions and therefore replacing general
\(c\) -concave functions with discrete \(c\) -concave functions leads to a
universal Riemannian OT model. Related, [5]} considered OT maps
of discrete measures which are defined via discrete \(c\) -concave functions.
